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A106234
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Triangle of the numbers of different forests with one or more isolated vertices. Those forests of rooted trees, have order N and m trees.
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2
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1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 9, 6, 3, 1, 1, 0, 20, 16, 7, 3, 1, 1, 0, 48, 37, 18, 7, 3, 1, 1, 0, 115, 96, 44, 19, 7, 3, 1, 1, 0, 286, 239, 117, 46, 19, 7, 3, 1, 1, 0, 719, 622, 299, 124, 47, 19, 7, 3, 1, 1, 0, 1842, 1607, 793, 320, 126, 47, 19
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| The unique tree with an isolated node has order one. For N>1 and m>1 there is at least one partition of N in m parts, with a part equal to 1, so a(n)>0, when m>1 and a(n) = 0, when m = 1 and N>1.
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FORMULA
| a(n)= sum over the partitions of N:1K1+2K2+ ... +NKN, with exactly m parts and one or more parts equal to 1, of product_{1=<i<=N}C(A000081[i]+Ki-1, Ki).
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EXAMPLE
| a(13) = 3 because 5 vertices can be partitioned in 3 trees in two ways: 1)one tree gets 3 nodes and the others get 1 each. 2)two trees get 2 nodes each and the other gets 1. Case 1) corresponds to 2 forests since A000081[3] = 2. Case 2) corresponds to one forest since A000081[2] = 1.
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CROSSREFS
| Cf. A033185, A105820.
Sequence in context: A147787 A135221 A191347 * A062507 A131044 A077875
Adjacent sequences: A106231 A106232 A106233 * A106235 A106236 A106237
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KEYWORD
| nonn,tabl
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AUTHOR
| Washington Bomfim (webonfim(AT)bol.com.br), Apr 26 2005
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